minimal infinite sigma algebra

Question

Does there exist sigma algebra whose cardinality is countably infinite? If yes tell me some examples. If not how to show every infinite sigma algebra is uncountable?

Answer 1

No countably infinite $\sigma$-algebras exist and here's a proof.

Let $\mathcal{A}$ be a $\sigma$-algebra with infinitely many members and $\mathcal{S}$ a subset of $\mathcal{A}$ with countably infinitely many members such that $\mathcal{S}$ covers the underlying space $\Omega$. Such a collection can always be found simply by adding the set $\{\Omega\}$ to any countably infinite collection of members of $\mathcal{A}$. Next we define a function $f \colon \Omega \to \mathcal{A}$ where $f(x)$ is the intersection of all members of $\mathcal{S}$ of which $x$ is a member. Since $\mathcal{S}$ is countably infinite, the intersection is countably infinite and thus $f(x)$ really does belong to $\mathcal{A}$. Clearly the function $f$ forms a partition of $\Omega$ and since every member $S$ of $\mathcal{S}$ is a union of the sets $f(x)$, $x \in S$, the partition has to be infinite. If the partition is uncountable, we are done, and if it is countably infinite, arbitrary unions of its members are all in $\mathcal{A}$, but there are uncountably many such unions.